Nobel Economists as Matchmakers in Medicine and Schools

How should New York City assign
students to schools? How should young doctors be matched with
residencies? How should lifesaving kidneys be allocated to
desperate patients?

For the answers, we have the work of the 2012 winners of
the Nobel Memorial Prize in Economic Sciences, Alvin Roth, a
Harvard colleague who is soon leaving for Stanford, and Lloyd Shapley, a professor emeritus at the University of California,
Los Angeles.

The world is full of allocation problems, and in most
cases, economists favor standard markets. How should we allocate
petroleum or Chevrolets or old-master paintings?

Economics 101 suggests that these goods should go to the
people who are willing to pay the most for them and therefore
value them most highly. Sometimes, as with gas stations, the
seller just posts a price and sells to anyone willing to pay.
Other times, as at a General Motors dealership, there is a bit
of haggling. The 1996 and 2007 Nobel prizes acknowledged the
pioneering work of William Vickrey and Roger Myerson in
designing auctions, but in all these settings the basic
objective remains the same: get the good to the person who is
willing to pay the most.

In the problems analyzed by Roth and Shapley, the right
answer isn’t to let the highest bidder prevail. In some
settings, there are two-sided preferences; residents care about
which hospital employs them, and hospitals care about which
residents they get.

Selling Spots

Harvard could sell its undergraduate spots to the highest
bidder, but this would reduce the quality of a Harvard
education. We don’t sell kidneys, because we are understandably
squeamish about auctioning off the right to live.

Shapley began this literature in 1962 with a paper co-
written with David Gale, who might have been included in this
prize if he hadn’t died in 2008. They focused on a particularly
ubiquitous matching problem, the marriage market, where the
standard economic answer, a cash auction, isn’t terribly
sensible.

A few medieval monarchs might have given their daughters to
the highest bidder, but it’s a good bet that if Catherine of
Aragon’s preferences had counted for anything, she would have
married someone other than Henry VIII. Stable marriages require
a matching mechanism that respects the preferences of both
spouses.

Gale and Shapley pondered both the marriage question, which
involved matching one man and one woman (this was 1962, after
all), and the college assignment problem, where multiple
applicants were sorted to a single choice. They produced a
matching algorithm that is surprisingly plausible, given the
authors’ interest in high abstraction.

The Gale-Shapley mechanism has men first propose to their
favorite woman. Women who receive more than one proposal string
along their favorite candidate and reject the others. The
rejected men make new proposals to any woman who has not
rejected them already, and the process continues. Whenever a
woman receives an offer she likes better than that from the man
she is stringing along, she drops her current beau and chooses
the new one.

Since the authors assume that the number of men and women
is equal and that everyone prefers marriage to solitude,
eventually everyone finds a mate. The allocation created by this
process is stable: No women or men would like to switch their
current spouse for someone else who would also prefer them to
their current spouse. The more desirable man would have proposed
to the woman first if he thought she was a more desirable mate.

Stable Outcomes

The allocation mechanism also works on Sadie Hawkins Day,
if the girls are doing the picking, and it also works for
college admissions. Each applicant applies to his or her
favorite college, and the college can decide to put the
applicant on a waiting list. The rejected applicants then apply
to other colleges, until the process sorts itself out. Not only
will the eventual outcome be stable, it will also be as good for
the applicants as any other stable allocation mechanism.

In 1982, however, Roth pointed out a problem with actually
carrying out the Gale-Shapley mechanism: People have an
incentive to lie. A woman may reject a man she actually likes
more, because that man, if rejected, will end up breaking up the
current understanding between some other woman and a man she
likes even better. For matching mechanism to work well, they
need to elicit honest behavior, not strategic play.

This truth-telling problem remains when we move from a
decentralized process of courtship to a centralized clearing-
house mechanism, as the medical internship market did in the
early 1950s. As Roth describes in a 1984 article, the
decentralized market for interns was painfully chaotic in the
1930s and 1940s. Interns would string hospitals along until the
last possible date, hoping that a better offer would show up,
and hospitals tried to eliminate this uncertainly by requiring
students to sign binding agreements early in their medical
school careers.

To regularize the market, a central clearing bureau was
established, which would elicit a list of preferences from both
the students and the hospitals, and then use a matching
algorithm to assign interns. The first proposed matching
algorithm had hospitals ranking students into five groups and
students ranking hospitals, and then matched the top choice of
one with the top choice of the other, and so on.

But in this design, students have an incentive to lie, and
overstate their desire to work in hospitals that are less likely
to fill up immediately. The system was dropped in favor of one
with “tentative matches” that can be undone if a better match
appears as the process unfolds, and that is closer to the Gale-
Shapley mechanism. While Roth proved that this system is
unlikely to elicit perfect truth-telling, it is far better than
the first proposal.

Good Match

Roth’s expertise made him a natural fit for New York City
to consult in 2003 when it was designing its school matching
system. Before 2003, New York’s schools allocated spots through
a complicated process involving repeated student applications,
rejections and much agony.

The system wasn’t only cumbersome. It also led to strategic
behavior, as when people applied first to less popular schools
to make sure they got in, leaving students with none of their
top choices. Roth pushed toward a clearing-house system, based
loosely on the models foreseen by Gale and Shapley and carried
out by hospitals. The result wasn’t exactly his ideal. It had
two rounds instead of one to address New York’s specialized exam
schools, but the new system seems to be a major step forward.

In recent years, Roth has been deeply involved in solving
the kidney-exchange problem. Distinguished economists have long
advocated a free market, where kidneys go to the highest bidder
and organ donors are paid handsomely. As Roth notes, however,
“repugnance” can be a powerful “constraint on markets.” We don’t
much like the idea of selling life itself, yet without financial
markets, allocating kidneys can be as tricky as allocating slots
in New York public schools.

The classic problem occurs when an ailing individual has a
prospective donor, but that donor’s kidney is immunologically
incompatible with the patient’s body. In a 2004 article, which I
edited in the Quarterly Journal of Economics, Roth and two co-
authors explore sensible kidney exchange systems. Roth proposes
a system, which has its roots in earlier work by Shapley and in
an idea from Gale.

The basic system is that each sick person, who has a
donated kidney that doesn’t work for them, points to their
preferred kidney in the exchange. This system must produce a
cycle, in which person A prefers kidney B and person with kidney
B prefers kidney C and person with kidney C prefers kidney A.
The algorithm works by satisfying the bargains implied by the
cycle, which must make everyone who trades better off, and
starting again. Roth’s algorithm is more complicated, but it
still hews to the basic goal of allocating kidneys fairly and in
a way that maximizes the number of lives saved.

Roth and Shapley have done much more than just design
exchanges. Shapley is the giant of cooperative game theory; Roth
is a pioneering experimental economist. Their work is very far
from the contentious debates of macroeconomics, and it
illustrates economics in action at its best. Not only scholars,
but children in New York City and people with kidney diseases
should be cheering for this year’s choices for the Nobel.

(Edward Glaeser, an economics professor at Harvard
University, is a Bloomberg View columnist. He is the author of
“Triumph of the City.” The opinions expressed are his own.)